Time Fractional Evolution Equations Research Articles

Memory approximate controllability properties for higher order Hilfer time-fractional evolution equations

Memory approximate controllability properties for higher order Hilfer time-fractional evolution equations

The backward problem for time-fractional evolution equations

In this paper, we consider the backward problem for fractional in time evolution equations with the Caputo derivative of order , where A is a self-adjoint and bounded above operator on a Hilbert space H. First, we extend the logarithmic convexity technique to the fractional framework by analyzing the properties of the Mittag–Leffler functions. Then we prove conditional stability estimates of Hölder type for initial conditions under a weaker norm of the final data. Finally, we give several applications to show the applicability of our abstract results.

A study of the wave dynamics of the space–time fractional nonlinear evolution equations of beta derivative using the improved Bernoulli sub-equation function approach

The space–time fractional nonlinear Klein-Gordon and modified regularized long-wave equations explain the dynamics of spinless ions and relativistic electrons in atom theory, long-wave dynamics in the ocean, like tsunamis and tidal waves, shallow water waves in coastal sea areas, and also modeling several nonlinear optical phenomena. In this study, the improved Bernoulli sub-equation function method has been used to generate some new and more universal closed-form traveling wave solutions of those equations in the sense of beta-derivative. Using the fractional complex wave transformation, the equations are converted into nonlinear differential equations. The achieved outcomes are further inclusive of successfully dealing with the aforementioned models. Some projecting solitons waveforms, including, kink, singular soliton, bell shape, anti-bell shape, and other types of solutions are displayed through a three-dimensional plotline, a plot of contour, and a 2D plot for definite parametric values. It is significant to note that all obtained solutions are verified as accurate by substituting the original equation in each case using the computational software, Maple. Additionally, the results have been compared with other existing results in the literature to show their uniqueness. The proposed technique is effective, computationally attractive, and trustworthy to establish more generalized wave solutions.

Exact Solutions to Some Nonlinear Time-Fractional Evolution Equations Using the Generalized Kudryashov Method in Mathematical Physics

In this study, we utilize the potent generalized Kudryashov method to address the intricate obstacles presented by fractional differential equations in the field of mathematical physics. Specifically, our focus centers on obtaining novel exact solutions for three pivotal equations: the time-fractional seventh-order Sawada-Kotera-Ito equation, the time-fractional Caudrey-Dodd-Gibbon-Sawada-Kotera equation, and the time-fractional seventh-order Kaup–Kupershmidt equation. The generalized Kudryashov method, celebrated for its versatility and efficacy in addressing intricate nonlinear problems, plays a central role in our research. This method not only simplifies the equations but also unveils their inner dynamics, rendering them amenable to meticulous analysis. It is worth noting that our fractional derivatives are defined in the context of the conformable fractional derivative, providing a solid foundation for our mathematical investigations. One notable aspect of our study is the visual representation of our findings. Graphical representations of the yielded solutions enliven intricate mathematical structures, providing a concrete insight into the dynamics and behaviors of said equations. This paper highlights the proficiency of the generalized Kudryashov method in resolving complex issues presented by fractional differential equations. Our study not only broadens the range of mathematical methods but also enhances our comprehension of the intriguing realm of nonlinear physical phenomena.

Decay estimates for the time-fractional evolution equations with time-dependent coefficients

In this paper, the initial-boundary value problems for the time-fractional degenerate evolution equations are considered. Firstly, in the linear case, we obtain the optimal rates of decay estimates of the solutions. The decay estimates are also established for the time-fractional evolution equations with nonlinear operators such as: p-Laplacian, the porous medium operator, degenerate operator, mean curvature operator and Kirchhoff operator. At the end, some applications of the obtained results are given to derive the decay estimates of global solutions for the time-fractional Fisher-KPP-type equation and the time-fractional porous medium equation with the nonlinear source.

(G'/G)-EXPANSION METHOD TO SEEK TRAVELING WAVE SOLUTIONS FOR SOME FRACTIONAL NONLINEAR PDES ARISING IN NATURAL SCIENCES

The (G'/G)-expansion method with the aid of symbolic computational system can be used to obtain the traveling wave solutions (hyperbolic, trigonometric and rational solutions) for nonlinear time-fractional evolution equations arising in mathematical physics and biology. In this work, we will process the analytical solutions of the time-fractional classical Boussinesq equation, the time-fractional Murray equation, and the space-time fractional Phi-four equation. With the fact that the method which we will propose in this paper is also a standard, direct and computerized method, the exact solutions for these equations are obtained.

The fractional evolution equations associated with the quantum fractional number operator

The Liouville–Caputo and Riemann–Liouville fractional derivatives have been two of the most useful operators for modeling nonlocal behaviors by fractional differential equations. In terms of Mittag–Leffler function and convolution product, using the Laplace transform, we give the exact values of the solutions of the Liouville–Caputo and Riemann–Liouville time fractional evolution equations associated with the Quantum fractional number operator. Therefore, we study the attractiveness of these solutions.

Uniqueness of orders and parameters in multi-term time-fractional diffusion equations by short-time behavior

As the most significant difference from parabolic equations, long-time or short-time behavior of solutions to time-fractional evolution equations is dominated by the fractional orders, whose unique determination has been frequently investigated in literature. Unlike all the existing results, in this article we prove the uniqueness of orders and parameters (up to a multiplier for the latter) only by principal terms of asymptotic expansions of solutions near t = 0 at a single spatial point. Moreover, we discover special conditions on unknown initial values or source terms for the coincidence of observation data. As a byproduct, we can even conclude the uniqueness for initial values or source terms by the same data. The proof relies on the asymptotic expansion after taking the Laplace transform and the completeness of generalized eigenfunctions.

Second-order nonuniform time-stepping schemes for time-fractional evolution equations with general elliptic operator

We extend the numerical methods in Lyu and Vong (2021) to the time-fractional evolution equations with general elliptic operator. By reformulating the mixed derivatives part of the elliptic operator and introducing a different auxiliary multiplicative function, the Alikhanov type schemes are proposed for the governing problems on general time meshes. Unconditional stability and second-order convergence are obtained by energy method.

On nonexistence of solutions to some time space fractional evolution equations with transformed space argument

Some results on nonexistence of nontrivial solutions to some time and space fractional differential evolution equations with transformed space argument are obtained via the nonlinear capacity method. The analysis is then used for a [Formula: see text] system of equations with transformed space arguments.

Subordination principle and Feynman-Kac formulae for generalized time-fractional evolution equations

We consider a class of generalized time-fractional evolution equations containing a fairly general memory kernel k and an operator L being the generator of a strongly continuous semigroup. We show that a subordination principle holds for such evolution equations and obtain Feynman-Kac formulae for solutions of these equations with the use of different stochastic processes, such as subordinate Markov processes and randomly scaled Gaussian processes. In particular, we obtain some Feynman-Kac formulae with generalized grey Brownian motion and other related self-similar processes with stationary increments.

Error estimate of the fast L1 method for time-fractional subdiffusion equations

This paper analyzes the convergence of the fast L1 method for the semi-linear time-factional subdiffusion equations. The tool we use is the generalized discrete Gronwall inequality. We show that the difference between the solution of the L1 method and the solution of the fast L1 method can be arbitrarily small and is independent of the sizes of the time and/or space grids. Our proof is very simple and is helpful to be used to simplify the convergence analysis of the fast methods for time-fractional evolution equations.

Uniqueness of inverse source problems for general evolution equations

In this paper, we investigate inverse source problems for a wide range of PDEs of parabolic and hyperbolic types as well as time-fractional evolution equations by partial interior observation. Restricting the source terms to the form of separated variables, we establish uniqueness results for simultaneously determining both temporal and spatial components without non-vanishing assumptions at [Formula: see text], which seems novel to the best of our knowledge. Remarkably, mostly we allow a rather flexible choice of the observation time not necessarily starting from [Formula: see text], which fits into various situations in practice. Our main approach is based on the combination of the Titchmarsh convolution theorem with unique continuation properties and time-analyticity of the PDEs under consideration.

Stochastic solutions of generalized time-fractional evolution equations

We consider a general class of integro-differential evolution equations which includes the governing equation of the generalized grey Brownian motion and the time- and space-fractional heat equation. We present a general relation between the parameters of the equation and the distribution of the underlying stochastic processes, as well as discuss different classes of processes providing stochastic solutions of these equations. For a subclass of evolution equations, containing Marichev-Saigo-Maeda time-fractional operators, we determine the parameters of the corresponding processes explicitly. Moreover, we explain how self-similar stochastic solutions with stationary increments can be obtained via linear fractional Lévy motion for suitable pseudo-differential operators in space.

Convergence analysis of the time-stepping numerical methods for time-fractional nonlinear subdiffusion equations

In 1986, Dixon and McKee (Z Angew Math Mech 66:535–544, 1986) developed a discrete fractional Gronwall inequality, which can be seen as a generalization of the classical discrete Gronwall inequality. However, this generalized discrete Gronwall inequality and its variant (Al-Maskari and Karaa in SIAM J Numer Anal 57:1524–1544, 2019) have not been widely applied in the numerical analysis of the time-stepping methods for the time-fractional evolution equations. The main purpose of this paper is to show how to apply the generalized discrete Gronwall inequality to prove the convergence of a class of time-stepping numerical methods for time-fractional nonlinear subdiffusion equations, including the popular fractional backward difference type methods of order one and two, and the fractional Crank-Nicolson type methods. We obtain the optimal \(L^2\) error estimate in space discretization for multi-dimensional problems. The convergence of the fast time-stepping numerical methods is also proved in a simple manner. The present work unifies the convergence analysis of several existing time-stepping schemes. Numerical examples are provided to verify the effectiveness of the present method.

Classical unique continuation property for multi-term time-fractional evolution equations

Classical unique continuation property for multi-term time-fractional evolution equations

Analytical solutions of the fifth-order time fractional nonlinear evolution equations by the unified method

This key purpose of this study is to investigate soliton solution of the fifth-order Sawada–Kotera and Caudrey–Dodd–Gibbon equations in the sense of time fractional local [Formula: see text]-derivatives. This important goal is achieved by employing the unified method. As a result, a number of dark and rational soliton solutions to the nonlinear model are retrieved. Some of the achieved solutions are illustrated graphically in order to fully understand their physical behavior. The results demonstrate that the presented approach is more effective in solving issues in mathematical physics and other fields.

Inverse problems for a multi-term time fractional evolution equation with an involution

This paper focuses on considering two inverse source problems (ISPs) for a multi-term time-fractional evolution equation with an involution term, interpolating the heat and wave equations. The fractional derivatives are defined in Caputo's sense. The ISPs are proved to be ill-posed in the sense of Hadamard. Recovering a space dependent source term from over-specified data given at some time constitute the first ISP, while in the second ISP determination of a time dependent component of the source term is considered when over-specified condition of integral type is given. The solution of ISPs are constructed by using Fourier's method. The time-dependent components of the solutions are presented in terms of the multinomial Mittag-Leffler function. Under certain conditions, the solutions of ISPs for the multi-term time-fractional evolution equation are shown to be classical solutions. In addition, some particular examples are formulated to illustrate the obtained results for the ISPs.

Second-Order and Nonuniform Time-Stepping Schemes for Time Fractional Evolution Equations with Time–Space Dependent Coefficients

The numerical analysis of time fractional evolution equations with the second-order elliptic operator including general time–space dependent variable coefficients is challenging, especially when the classical weak initial singularities are taken into account. In this paper, we introduce a concise technique to construct efficient time-stepping schemes with variable time step sizes for two-dimensional time fractional sub-diffusion and diffusion-wave equations with general time–space dependent variable coefficients. By means of the novel technique, the nonuniform Alikhanov type schemes are constructed and analyzed for the sub-diffusion and diffusion-wave problems. For the diffusion-wave problem, our scheme is constructed by employing the recently established symmetric fractional-order reduction method. The unconditional stability of proposed schemes is rigorously discussed under mild assumptions on variable coefficients and, based on reasonable regularity assumptions and weak time mesh restrictions, the second-order convergence is obtained with respect to discrete \(H^1\)-norm. Numerical experiments are given to demonstrate the theoretical statements.

Error estimates of a finite element method for stochastic time-fractional evolution equations with fractional Brownian motion

The aim of this paper is to consider the convergence of the numerical methods for stochastic time-fractional evolution equations driven by fractional Brownian motion. The spatial and temporal regularity of the mild solution is given. The numerical scheme approximates the problem in space by the Galerkin finite element method and in time by the backward Euler convolution quadrature formula, and the noise by the [Formula: see text]-projection. The strong convergence error estimates for both semi-discrete and fully discrete schemes are established. A numerical example is presented to verify our theoretical analysis.